Strict and pointwise convergence of BV functions in metric spaces

TL;DR

This paper proves that in metric spaces with certain properties, strict BV convergence implies pointwise convergence of approximate limits outside a negligible set.

Contribution

It establishes pointwise convergence of BV functions in metric spaces under strict convergence, extending classical results to a broader setting.

Findings

Strict convergence in BV implies pointwise convergence outside a negligible set.

The result holds in metric spaces with doubling measure supporting a Poincaré inequality.

Convergence of total variation norms is crucial for the pointwise convergence.

Abstract

In the setting of a metric space $X$ equipped with a doubling measure that supports a Poincar\'e inequality, we show that if $u_i\to u$ strictly in $BV(X)$, i.e. if $u_i\to u$ in $L^1(X)$ and $\Vert Du_i\Vert(X)\to\Vert Du\Vert(X)$, then for a subsequence (not relabeled) we have $\widetilde{u}_i(x)\to \widetilde{u}(x)$ for $\mathcal H$-almost every $x\in X\setminus S_u$.